Rigidity, Volume, and Combinatorics in Hyperbolic Geometry

在二十世纪的大部分时间里，描述三维空间形状的挑战被视为一个代数问题：实际上，我们首先通过代数区分了球体的(拓扑)结构和甜甜圈的结构。威廉·瑟斯顿的作品使这种空间的几何形状更清晰地进入人们的视野，成为探索的中心特征。这种方法的胜利是佩雷尔曼证明了瑟斯顿的几何化猜想，即每个三维流形都可以自然地分解成碎片，每个碎片都有统一的几何形状。通过刚性的概念，对这些几何的研究常常被简化为考虑由曲面上的环引起的简单组合结构。这项拟议的研究将探索这些结构如何预测与量子物理概念有关的体积、直径、长度和几何的其他方面。双曲三维流形的体积与Weil-Petersson距离之间的长期联系依赖于通过裤子图的组合比较，该图组织了曲面上的最大多曲线，而其他联系则来自Witten的工作通过重整化体积。Schlenker最近的工作在拟Fuchsian流形的背景下明确了这一联系，Pi提出的工作将进一步发展这一概念，明确地将纤化的3-流形和平移流形联系起来。更广泛地说，所提出的研究中的一个主要项目是巩固和扩展闭流形中组合学和几何之间的联系，并用这些工具继续研究Klein群的形变空间的结构。作为这些技术力量的一个例子，随机Heegaard分裂的双Lipschitz模型提供了一个完整的解决方案(与Rivin和Souto一起)，以解答Dunfield和瑟斯顿的猜想，即随机Heegaard分裂几乎肯定是双曲线的，其体积呈线性增长。最后，PI将与Minsky以及MoDami、Leininger和Rafi一起开展项目，研究组合数学在Weil-Petersson度量的测地线几何中的作用，调查TeichMuller和Weil-Petersson测地线在唯一遍历性方面的差异。

For much of the twentieth century, the challenge to describe the shape of three dimensional spaces was viewed as an algebraic problem: indeed it is through algebra that we first distinguish the (topological) structure of a sphere from that of a doughnut. Work of William Thurston brought the geometry of such spaces more clearly into view as a central feature to explore. The triumph of this approach was Perelman's proof of Thurston's geometrization conjecture, that each three-dimensional manifold could be naturally broken up into pieces, each with a uniform geometry. The study of these geometries is frequently reduced, via a notion of rigidity, to considering simple combinatorial structures arising from loops on surfaces. The proposed research will explore how these structures predict volume, diameter, length and other aspects of the geometry, relating to notions from quantum physics.A longstanding connection between volume of hyperbolic three-manifolds and Weil-Petersson distance relied on a combinatorial comparison via the pants graph which organizes maximal multicurves on a surface, yet other connections were known to arise from work of Witten via renormalized volume. Recent work of Schlenker made this connection explicit in the context of quasi-Fuchsian manifolds, and PI's proposed work will develop this idea further to explicitly relate fibered 3-manifolds and translation distiance. More generally, a primary project in the proposed research is to solidify and extend the connection between combinatorics and geometry in closed manifolds, and to continue to investigate the structure of deformation spaces of Kleinian groups with these tools. As an example of the power of these techniques, bi-Lipschitz models for 'random' Heegaard splittings provide a full solution (with Rivin and Souto) to the conjecture of Dunfield and Thurston that random Heegaard splittings are almost surely hyperbolic and their volume grows linearly. Finally, the PI will pursue projects with Minsky, and with Modami, Leininger and Rafi on the role of combinatorics in the geometry of geodesics in the Weil-Petersson metric, investigating disparities between Teichmuller and Weil-Petersson geodesics in terms of unique ergodicity.